ny times 23 sept 2008 - time series of the day. stat 153 - 23 sept 2008 d. r. brillinger chapter 4 -...
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Stat 153 - 23 Sept 2008 D. R. Brillinger
Chapter 4 - Fitting t.s. models in the time domain
sample autocovariance coefficient.
Under stationarity, ...
)/1(),( NOccCov mk
NaskcE k )()(
,...2,1,0 ,/))((1
kNxxxxc kt
kN
ttk
Estimated autocorrelation coefficient
0/ ccr kk
large and .. ,..., If 1 Ndiixx N
NrE k /1)(
NrVar k /1)(
asymptotically normal
interpretation
N/2 :0 around CI 95% eapproximat
Uses of acf
mixing (asymptotically independent)?
MA(q)?
Seasonal component?
ergodic
N
tt NXfENXf
1 asy probabilitin ))((/)(
Estimating the mean
N
tt NXX
1/
unbiased ,)( XE
1
1
2 /)]()1(21[)(N
rNr
Nr
XVar
Can be bigger or less than 2/N
NAR /)11
(approx )1(For 2
Fitting an autoregressive, AR(p)
Easy. Remember regression and least squares
nixY iii ,...,1 ,
2
),( ][min ii xy
normal equations
iiii
ii
yxxx
yx
ˆ ˆ
ˆ ˆ12
... ' ,)(...)( 11 diisZZXXX tptptt
N
ptptptt xxxS
1
2
11 )](...)([
AR(1)
1
1
1
1
2
1 )(/))((ˆN
t
N
tttt xxxxxx
Cp.
011 / ccr
Fitting an MA(q).
Later. There is an R program
Fitting an ARMA(p,q).
Later. There is an R program
Estimating p, q, (p,q).
Later. There is a criterion.
Seasonal ARIMA. seasonal parameter s
SARIMA(p,d,q)(P,D,Q)s
Example
tt ZBXBB )1()1)(1( 1212
1213112 )( tttttt ZZXXXX
t
s
Qqt
s
Pp ZBBWBB )()()()(
Residual analysis.
Paradigm
observation = fitted value plus residual
The parametric models have contained Zt
tz is residual The
1t ˆz
AR(1).
tt xx
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